You can open older projects in newer engine versions easily, but not the other way around.
I.e. You can't downgrade. Any asset created or saved with a newer version won't be recognized by an older editor (the project will open, it just won't show any assets).
As for the pulldown in your screenshot, you are adding an asset that isn't listed for your project's engine version. E.g. your project is 4.26 while the asset is 4.22-4.25, as an example.
Oftentimes, your choice won't make a difference as most assets on the marketplace are maintained in the lowest possible version and made compatible with newer ones just by flipping a checkbox in the project page.
However, some assets will be available in several variants (e.g. different files for 4.22, yet different ones for 4.23, etc. You can see this with the free Advanced Locomotion System, just as an example, where V1, V2, V3 and V4 coexist). In such a case your selection will download the corresponding version.
You usually don't know which is the case. So selecting the closest makes most sense.
It selects the version of UE to use. The projects say which version was used to create them (lower right corner of project).
Will the project downgrade to that version?
I assume no.
You can open older projects in newer engine versions easily, but not the other way around. I.e. You can't downgrade. Any asset created or saved with a newer version won't be recognized by an older editor (the project will open, it just won't show any assets). As for the pulldown in your screenshot, you are adding an asset that isn't listed for your project's engine version. E.g. your project is 4.26 while the asset is 4.22-4.25, as an example. Oftentimes, your choice won't make a difference as most assets on the marketplace are maintained in the lowest possible version and made compatible with newer ones just by flipping a checkbox in the project page. However, some assets will be available in several variants (e.g. different files for 4.22, yet different ones for 4.23, etc. You can see this with the free Advanced Locomotion System, just as an example, where V1, V2, V3 and V4 coexist). In such a case your selection will download the corresponding version. You usually don't know which is the case. So selecting the closest makes most sense.